**3. Results and Discussion**

The nonlinear ordinary differential Equations (12) and (13) subject to boundary restrictions (Equation (14)) were solved numerically using the bvp4c function in the MATLAB software. The effects of various physical parameters (the magnetic parameter *M*, curvature parameter *B*, slip parameter *B*2, nanoparticle volume fraction *φ*, radiation parameter *Rd*, suction *S*, and stretching/shrinking parameter *λ*) on the velocity profile, temperature distribution, skin friction coefficient, and heat transfer rate are illustrated in Figures 2–19 for both the carrier-based fluid (*φ* = 0) and the gold particle nanofluid (*φ* = 0.035). The parameters fixed throughout computation were *B*1 = 1.1, *S* = 3.5 , *M* = 0.5, *B*2 = 0.2, *λ* = 1.5,*n* = *Rd* = 2, and *B* = 1.5.

**Figure 2.** Effect of a magnetic field on the velocity profile.

**Figure 3.** Effect of a partial slip on the velocity profile.

**Figure 4.** Effect of suction on the velocity profile.

**Figure 5.** Effect of the radius of curvature on the velocity profile.

**Figure 6.** Effect of the nanoparticle volume fraction on the velocity profile.

**Figure 7.** Effect of the material parameter on the velocity profile.

**Figure 8.** Effect of stretching/shrinking on the velocity profile.

**Figure 9.** Effect of the magnetic field on the temperature distribution.

**Figure 10.** Effect of the slip on the temperature distribution.

**Figure 11.** Effect of suction on the temperature distribution.

**Figure 12.** Effect of stretching/shrinking on the temperature distribution.

**Figure 13.** Effect of curvature on the temperature distribution.

**Figure 14.** Effect of radiation on the temperature distribution.

**Figure 15.** Effect of the nanoparticle volume fraction on the temperature distribution.

**Figure 16.** Effect of the slip on the skin friction coefficient against stretching/shrinking.

**Figure 17.** Effect of the slip on the Nusselt number against stretching/shrinking.

**Figure 18.** Effect of suction on the skin friction coefficient against stretching/shrinking.

**Figure 19.** Effect of suction on the Nusselt number against stretching/shrinking.

### *3.1. Effect of Physical Parameters on the Velocity Profile*

Figures 2–6 demonstrate the behavior of the velocity profile at different values of *M*,*B*2,*S*,*B*, and nanoparticle volume fraction *φ*. The velocity decreases due to the magnetic field for the carrier-based fluid and the gold particle nanofluid (Figure 2). Initially, the velocity and thickness of the momentum boundary layer for the carrier-based fluid are larger compared to those for the gold particle nanofluid. In addition, the velocity behavior of the gold particle nanofluid reduces more in the presence of gold nanoparticles, because gold nanoparticles generate friction in the fluid. Physically, increasing values of *M* augmen<sup>t</sup> the Lorentz force, which ultimately decreases velocity. Figure 2 also shows that the velocity of blood decreases. Figure 3 shows the effect of a partial slip on the velocity. The velocity of the fluid decreases for both the carrier-based fluid and the gold particle nanofluid. The velocity decreases in both fluids (*φ* = 0) and (*φ* = 0.035) with increasing *B*2. The velocity reduction at the curve surface shows that the fluid flow occurs at the stretching curve surface; thus, any increase in the velocity slip parameter of the fluid at the stretching surface decreases the velocity field. Moreover, the higher values of *B*2 signify that the friction connecting the blood and the surface is removed. The impact of suction on the velocity is shown in Figure 4 for both the carrier-based fluid and the gold particle nanofluid. Physically, the resistance in the blood flow occurs because of viscosity, which can be handled by using suction. Increasing suction decreases the drag force at the sheet. Consequently, the momentum boundary layer thickness also reduces in both cases.

Figure 5 shows the effect of the curvature on the velocity profile, which increases the flow for both the carrier-based fluid and the gold particle nanofluid. The momentum boundary layer thickness and the magnitude of the velocity also improve with increasing curvature. Furthermore, the distance between the solution curves for the carrier-based fluid is slightly similar compared to that between the outcome curves of the gold particle nanofluid. Generally, this behavior of a fluid means that the bend of the curved stretching surface enhances fluid flow over it. This rise in the velocity gradient is slightly more in the carrier-based fluid compared to the gold particle nanofluid.

Figure 6 shows that the velocity of blood decreases as *φ* increases, which is responsible for the reduction in the velocity of the boundary layer. Physically, a higher *φ* enhances the blood viscosity, which consequently decreases the magnitude of the boundary layer thickness. Figures 7 and 8 show that the velocity increases for both the carrier-based fluid and the gold particle nanofluid due to the material and stretching/shrinking parameters. In addition, the magnitude of the velocity and the momentum boundary layer thickness increase with increasing material and stretching parameters. Both graphs are plotted for

both fluids, where the increase in velocity is more for the carrier-based fluid compared to the gold particle nanofluid.

### *3.2. Effect of Physical Parameters on the Temperature Distribution*

Figures 9–15 show the effects of the magnetic, slip, suction, stretching/shrinking, curvature, and radiation parameters on the temperature for the carrier-based fluid and the gold particle nanofluid. Figure 9 shows that the temperature increases due to the magnetic field for both fluids. This is because inclusion of the transverse strength of a magnetic field in an electrically conducting fluid increases the Lorentz force. This strength bears the potential to increase the blood temperature distribution.

Figure 10 shows that the velocity slip increases the temperature of the blood. The temperature distribution and the thermal boundary layer thickness increase with increasing slip for both the carrier-based fluid and the gold particle nanofluid because the slip slows down the fluid motion and ultimately affects the temperature. It is also evident from this figure that the increase in temperature is more for the nanofluid as compared to the regular fluid for the larger impacts of slip constraint.

Figure 11 shows that for larger values of suction, the blood temperature at any point of the flow is moderate for both the carrier-based fluid and the gold particle nanofluid. The thermal boundary layer thickness and the temperature distribution decrease at higher suction for both fluids. Stretching/shrinking reduces the temperature, as shown in Figure 12, for both fluids. A similar behavior is seen in Figure 12 for both fluids due to larger stretching compared to Figures 11 and 13, and shows that the temperature reduces due to the curvature for both fluids. The thermal boundary layer thickness and the temperature distribution decrease for both fluids due to the higher curvature. The figure insert shows the blood temperature distribution in terms of the significant effect of curvature, where the thermal conductivity is more for the gold particle nanofluid compared with the carrier-based fluid. From a physical point of view, this happens because the stretching curved surface increases the fluid flow for the velocity profiles and indirectly contributes to reducing the temperature distribution in terms of magnitude.

Radiation increases the temperature for both the carrier-based fluid and the gold particle nanofluid (Figure 14), which consequently increases the significant boundary layer thickness. Thus, the temperature of the boundary layer increases significantly. Moreover, the temperature distribution is greater for the gold particle nanofluid compared with the carrier-based fluid because the presence of gold nanoparticles produces more energy in the form of heat and, consequently, the temperature rises. The temperature increases due to the nanoparticle volume fraction in both fluids (Figure 15), because the inclusion of gold nanoparticles increases the thermal conductivity of blood, which increases the blood temperature.

### *3.3. Effect of Physical Parameters on the Skin Friction Coefficient and the Nusselt Number*

The effects of the slip on skin friction coefficient and the Nusselt number against stretching/shrinking ( *λ*) for both the carrier-based fluid and the gold particle nanofluid are shown in Figures 16 and 17, respectively. The velocity slip enhances the skin friction coefficient but reduces the Nusselt number in both fluids. In addition, the skin friction coefficient significantly shrinks due to stretching/shrinking, whereas heat transfer increases. Moreover, the Nusselt number is higher in the gold particle nanofluid compared to the carrier-based fluid, which ultimately enhances thermal conductivity. The effects of suction on shear stress and heat transfer against *λ* are shown in Figures 18 and 19, respectively. The skin friction coefficient shrinks due to suction, whereas the Nusselt number increases. The thermophysical data of the base fluid (blood) and gold nanoparticles are listed in Table 1.


**Table 1.** Thermophysical properties of blood and gold nanoparticles (Koriko et al. [32]).

Finally, Table 2 presents a comparison of the current outcomes of the skin friction coefficient for *B* when *B*1 = *Rd* = *S* = *M* = *B*2 = *φ* = 0, *λ* = 1, and *n* = 1, which shows favorable agreement. For more details about the current technique, Table 3 shows a comparison of the current computational outcomes of the shear stress or friction factor for distinct values of the shrinking parameter when *B* → <sup>∞</sup>, *B*2 = *M* = *φ* = *B*1 = 0, *n* = 1, and *S* = 2 with the results of Ro¸sca et al. [33]. The values show excellent agreement, proving the feasibility of the present numerical scheme. In addition, the numerical computational values of the skin friction coefficient and the heat transfer rate for the various constraints are given in Table 4 for both the carrier-based fluid (*φ* = 0) and the gold particle nanofluid (*φ* = 0.035), while the rest of the fixed parameters are Pr = 21, *n* = 2, *S* = 3.5, and *B*2 = 0.2. For the carrier-based fluid, the skin friction coefficient increases by 18.815%, 27.626%, 6.231%, and 1.552 × 10−4% due to the impact of *λ*, *M*, *B*1, and *Rd*, respectively, while it decreases by 1.774% due to *B*. By contrast, for the gold particle nanofluid, the skin friction coefficient increases by 18.311%, 15.034%, 4.541%, and 3.207 × 10−5% due to the effect of *λ*, *M*, *B*1, and *Rd*, respectively, while it decreases by approximately 0.751% due to the curvature parameter. Moreover, due to *λ* and *B*1, the heat transport rate increases by 0.157% and 0.031%, respectively, for the carrier-based fluid and by 0.156% and 0.027%, respectively, for the gold particle nanofluid. Due to the effect of the magnetic, curvature, and radiation parameters, the heat transport rate decreases by 0.078%, 6.42 × <sup>10</sup>−3%, and 41.705%, respectively, for the carrier-based fluid and by 0.053%, 8.39 × <sup>10</sup>−3%, and 41.045%, respectively, for the gold particle nanofluid. Generally, the increasing skin friction coefficient is better for both fluids due to the magnetic and stretching parameters, while it is lower (approximately 1.552 × 10−4% and 3.207 × <sup>10</sup>−5%, respectively) for the radiation parameter. Alternatively, the heat transport rate is maximum for the stretching parameter for both fluids when the parameter increases and minimum (about 6.42 × 10−3% and 8.39 × <sup>10</sup>−3%, respectively) for the curvature parameter. Finally, these numerically calculated values show that the skin friction coefficient and the heat transfer rate are largely found in the carrier-based fluid compared to the gold particle nanofluid.

**Table 2.** Comparison of the skin friction coefficient −1 2Re*<sup>b</sup>* 1 *n*+1 *CF* for different values of *B* with the results of [34–36].



**Table 3.** Comparison of the skin friction coefficient −1 2Re*<sup>b</sup>* 1 *n*+1 *CF* for different values of *λ* with the results of Ro¸sca et al. [33].

**Table 4.** Skin friction coefficient 1 2 Re*b* 1 *n*+1 *CF* and Nusselt number Re*b* −1 *n*+1 *Nus* for different parameters at Pr = 21, *n* = 2, *S* = 3.5, and *B*2 = 0.2.

